On convergence of numerical solutions for the compressible MHD system with weakly divergence-free magnetic field

TL;DR

This paper develops a convergence theory for numerical solutions of the compressible MHD system, introducing dissipative weak solutions and proving their uniqueness and convergence properties, with applications to specific numerical methods.

Contribution

It introduces a new convergence framework for compressible MHD equations, including dissipative weak solutions and a nonlinear Lax-equivalence theorem variant.

Findings

Proved weak-strong uniqueness for dissipative weak solutions.

Established convergence of numerical approximations to weak and classical solutions.

Applied the theory to analyze two numerical methods.

Abstract

We study a general convergence theory for the analysis of numerical solutions to the magnetohydrodynamic system describing the time evolution of compressible, viscous, electrically conducting fluids in space dimension d (= 2; 3). First, we introduce the concept of dissipative weak solutions and prove the weak-strong uniqueness property for dissipative weak solutions, meaning a dissipative weak solution coincides with a classical solution emanating from the same initial data on the lifespan of the latter. Next, we introduce the concept of consistent approximations and prove the convergence of consistent approximations towards the dissipative weak solution as well as the classical solution. Interpreting the consistent approximation as the energy stability and consistency of numerical solutions, we have built a nonlinear variant of the celebrated Lax-equivalence theorem. Finally, as an application of this theory, we show the convergence analysis for two numerical methods.